Optimal. Leaf size=29 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0203407, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {446, 76} \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 76
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a A}{x^3}+\frac{A b+a B}{x^2}+\frac{b B}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a A}{4 x^4}-\frac{A b+a B}{2 x^2}+b B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0171773, size = 31, normalized size = 1.07 \[ \frac{-a B-A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 28, normalized size = 1. \begin{align*} bB\ln \left ( x \right ) -{\frac{Aa}{4\,{x}^{4}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{Ba}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01377, size = 41, normalized size = 1.41 \begin{align*} \frac{1}{2} \, B b \log \left (x^{2}\right ) - \frac{2 \,{\left (B a + A b\right )} x^{2} + A a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46829, size = 73, normalized size = 2.52 \begin{align*} \frac{4 \, B b x^{4} \log \left (x\right ) - 2 \,{\left (B a + A b\right )} x^{2} - A a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.47491, size = 27, normalized size = 0.93 \begin{align*} B b \log{\left (x \right )} - \frac{A a + x^{2} \left (2 A b + 2 B a\right )}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13303, size = 53, normalized size = 1.83 \begin{align*} \frac{1}{2} \, B b \log \left (x^{2}\right ) - \frac{3 \, B b x^{4} + 2 \, B a x^{2} + 2 \, A b x^{2} + A a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]